Transmission Line Reflectionless Filters

ABSTRACT

Reflectionless transmission line filters, as well as a method for designing such filters is disclosed. These filters preferably function by absorbing the stop-band portion of the spectrum rather than reflecting it back to the source, which has significant advantages in many different applications. The insertion of additional transmission line sections that change the phase response of the circuit without altering the amplitude response preferably allows follow-up transmission line identities to be applied in order to arrive at a more easily manufacturable filter topology. This facilitates their application over a higher frequency range the solely lumped-element circuits.

REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. Non-Provisional applicationSer. No. 14/927,881, filed Oct. 30, 2015, which claims priority to U.S.Provisional Application No. 62/075,499, filed Nov. 5, 2014, both ofwhich are entitled “Transmission Line Reflectionless Filters,” and arehereby specifically and entirely incorporated by reference.

RIGHTS IN THE INVENTION

This invention was made with government support under CooperativeAgreement AST-0223851, between the National Science Foundation andAssociated Universities, Inc., and, accordingly, the United Statesgovernment has certain rights in this invention.

BACKGROUND 1. Field

The invention is directed toward electronic filters and methods of theiruse. Specifically, the invention is directed toward reflectionlesselectronic filters and methods of their use.

2. Background

Virtually all electronic systems use some kind of filtering to rejectunwanted frequency components. In most conventional filters, therejected signals are bounced back to the source, eventually dissipatingin the generator itself, or in the interconnecting wires/transmissionlines, or being radiated into the instrument housing. This manner ofrejecting unwanted signals can sometimes lead to harmful interactionswith other components in the system, either by spurious mixing innon-linear devices, unintentional re-biasing of sensitive activecomponents, or cross-talk between various signal paths. A solution wassought for a filter that would absorb these unwanted signals before theycould compromise performance. This led to a novel absorptive filtertopology which was patented in 2013 (U.S. Pat. No. 8,392,495) as well aspending U.S. patent application Ser. No. 14/724,976 the entirety of bothof which are incorporated by reference herein. These absorptive filterssolved many problems encountered with conventional filters, such as thesensitivity of mixers to poor out-of-band terminations, detrimental anddifficult-to-predict non-linear effects from reactive harmonic loading,leakage or cross-talk due to trapped energy between the filter and otherpoorly-matched components, and numerous other problems associated without-of-band impedance matching. They also realized superior performanceand manufacturability when compared to other approaches to absorptivefilters, such as terminated diplexers and directional filter structuresemploying quadrature hybrids.

None of these prior embodiments, however, adequately taught how toimplement such designs using transmission lines rather than lumpedelements. The inability to easily convert these into transmission lineform constrained the frequencies at which they could be effectivelyimplemented to the cm-wave range and below. Recent efforts to addressthis issue have yielded a practical transmission line solution whichextends the frequency range easily into submillimeter-waves whilemaintaining the benefits of the original reflectionless filter topology.

SUMMARY

The present invention addresses several of the problems anddisadvantages associated with conventional filters, and with the priorart of reflectionless filters, thereby providing a new resource for bandselection and definition in electronic systems.

An embodiment of the invention is directed to a reflectionlesselectronic filter. The filter comprises a symmetric two-port circuit,wherein the symmetry defines an even-mode equivalent circuit and anodd-mode equivalent circuit when the ports are driven in-phase and 180°out-of-phase, respectively; at least one transmission line and at leastone lossy element or matched internal sub-network arranged within thesymmetric two-port circuit such that: a normalized input impedance ofthe even-mode equivalent circuit is substantially equal to a normalizedinput admittance of the odd-mode equivalent circuit and a normalizedinput impedance of the odd-mode equivalent circuit is substantiallyequal to a normalized input admittance of the even-mode equivalentcircuit. In this way, the even- and odd-mode equivalent circuits aresaid to be duals of each other.

To design a reflectionless filter using transmission lines, one maystart with a lumped element prototype. Richard's Transformations arethen applied to convert the circuit as derived into a transmission lineform, called a network. Finally, transmission line identities are usedto modify the transmission line network so that it is easier tomanufacture in a desired medium (such as microstrip, coplanar waveguide,stripline, coaxial cable, waveguide, etc.) Application of theseidentities are often facilitated by the introduction of matchedtransmission line sections in cascade with the intermediate filterstructures while the filter topology is being developed. This allowscertain identity transforms to be used that would not be possible withthe intrinsic filter structures.

In some embodiments, the resulting network comprises resonators formedby transmission-line stubs separated by cascaded transmission lines. Inother embodiments, the resonators may be formed by alternating high- andlow-impedance transmission lines instead of, or in combination with, thetransmission line stubs.

Note that throughout this document, the terms “parallel” and “shunt”will be used synonymously with regard to the connection of circuitelements or transmission line stubs.

DESCRIPTION OF THE DRAWINGS

The invention is described in greater detail by way of example only andwith reference to the attached drawings, in which:

FIG. 1. Even-mode equivalent circuit of a prototype reflectionlessfilter using lumped elements with an arbitrary length of matchedcascaded transmission line at the input. The order of the filter at thisstage is arbitrary.

FIG. 2. Even-mode (left) and odd-mode (right) equivalent circuits of areflectionless filter in the early stages of derivation.

FIG. 3. Even- and odd-mode equivalent circuits incorporating seriesshort-circuited stubs and shunt open-circuited stubs after applicationof Richard's Transformations.

FIG. 4. Even- and odd-mode equivalent circuits after application ofKuroda's Identities.

FIG. 5. Even- and odd-mode equivalent circuits after swapping positionsof the series stub and termination resistor in the even-mode equivalentcircuit.

FIG. 6. Even- and odd-mode equivalent circuits after restoration ofsymmetry conditions near the termination resistors.

FIG. 7. A full two-port band-pass reflectionless filter usingtransmission lines.

FIG. 8. A transmission line identity that is useful for replacingseries-connected open-circuited stubs in the transmission linereflectionless filters with coupled transmission lines.

FIG. 9. The reflectionless filter of FIG. 7 after two applications ofthe identity in FIG. 8.

FIG. 10. A conventional directional filter for comparison with thereflectionless filter in FIG. 9.

FIG. 11. One embodiment of the transmission line reflectionless filterusing only a single pair of open-circuited stubs.

FIG. 12. A plot of the transfer characteristic of the reflectionlessfilter in FIG. 11.

FIG. 13. Alternate form of the reflectionless filter in FIG. 11. Thetransfer characteristics of the two forms are identical (shown in FIG.12) but with different coupled-line impedance parameters that may beeasier to fabricate in some cases.

FIG. 14. Another embodiment of the transmission line reflectionlessfilter using a cascaded line resonator instead of stubs.

FIG. 15. A plot of the transfer characteristic of the reflectionlessfilter in FIG. 14.

DETAILED DESCRIPTION

As embodied and broadly described herein, the disclosures herein providedetailed embodiments of the invention. However, the disclosedembodiments are merely exemplary of the invention that can be embodiedin various and alternative forms. Therefore, there is no intent thatspecific structural and functional details should be limiting, butrather the intention is that they provide a basis for the claims and asa representative basis for teaching one skilled in the art to variouslyemploy the present invention.

A problem in the art capable of being solved by the embodiments of thepresent invention is a circuit topology and design technique forelectronic filters that are well-matched at all frequencies. It has beensurprisingly discovered that such filters have a number of unexpectedadvantages, including minimal reflections on their input and outputports, either in their pass bands or stop bands, or the transitionbands. The return loss for these filters is substantially infinite atall frequencies. In conventional filters, on the other hand, stop bandrejection is achieved by reflecting the unwanted portion of the spectrumback toward the signal source rather than absorbing it. The instantfilters are comprised of transmission lines along with lumped elementresistors, inductors, and capacitors, or of transmission lineequivalents, and can be implemented in whatever form is suited to theapplication (e.g. waveguide, coaxial, wire-leaded, surface-mount,monolithically-integrated).

Initially, one starts with an arbitrary, symmetric, two-port network.While symmetry is not required of reflectionless filters, the preferredembodiment is symmetrical. In such a network, if both ports are excitedsimultaneously with equal signal amplitudes and matching phase, therewill be no currents crossing from one side of the symmetry plane to theother. This is called the even-mode. Similarly, if the two ports areexcited with equal amplitudes but 180° out of phase, then all nodes thatlie on the symmetry plane should have zero electric potential withrespect to ground. This is called the odd-mode.

Therefore, it is possible to have two single-port networks, eachcontaining one half of the elements of the original two-port network,where the nodes that lie on the symmetry plane are either open-circuitedor shorted to ground. These can be called the even-mode equivalentcircuit and the odd-mode equivalent circuit, respectively. Equivalentcircuits are circuits that retain all of the electrical characteristicsof the original (and often more complex) circuits. The scatteringparameters of the original two-port network are then given as thesuperposition of the reflection coefficients of the even- and odd-modeequivalent circuits, as follows:

s ₁₁ =s ₂₂=½(Γ_(even)+Γ_(odd))  (1)

s ₂₁ =s ₁₂=½(Γ_(even) −ΓF _(odd))  (2)

wherein s_(ij) is the scattering coefficient from port j to port i, andΓ_(even) and Γ_(odd) are the reflection coefficients of the even- andodd-mode equivalent circuits, respectively. Thus, the condition forperfect input match, s₁₁=0, is derived from (1) as follows:

Γ_(even)=−Γ_(odd)  (3)

This is equivalent to saying that the normalized even-mode inputimpedance is equal to the normalized odd-mode input admittance (orvice-versa):

z _(even) =y _(odd)  (4)

wherein z_(even) is the normalized even-mode impedance, and y_(odd) isthe normalized odd-mode admittance, which is satisfied if the even- andodd-mode circuits are duals of each other (e.g. inductors are replacedwith capacitors, shunt connections with series connections). Further, bycombining (2) and (3), the transfer function of the original two-portnetwork is given directly by the even-mode reflection coefficient:

s ₂₁=Γ_(even)  (5)

It is therefore often useful to construct the even-mode equivalentcircuit as the dual of the odd-mode equivalent circuit, and vice-versa.When the filter comprises transmission lines, the dual may beconstructed by replacing cascaded transmission lines with others havingthe inverse normalized characteristic impedance, and by replacingopen-circuited stubs with short-circuited stubs, andparallel-connections with series-connections. In some embodiments, itmay be necessary to apply transmission line identities to restoresymmetry after constructing the even- and odd-mode equivalent circuits,or to make the topology more easily manufacturable. In a preferredembodiment, the Kuroda Identities are especially useful to transformseries-connected stubs into parallel-connected stubs, or vice-versa.Note that to make this particular identity transformation possible, itis often useful to insert one or more matched cascade transmission linesat the input and/or at a lossy termination of the even- or odd-modeequivalent circuits.

In some preferred embodiments, it is useful to apply a transmission lineidentity that replaces a cascade transmission line having a transmissionline stub at one end with a coupled-transmission line. In otherembodiments, a series-connected stub may exchange positions with a lossytermination connected in series with it, resulting in a series lossyelement followed by a parallel-connected stub.

Note that reflectionless filters comprising transmission lines may beenhanced with matched-internal sub-networks. These sub-networks maythemselves comprise transmission lines, lumped-elements, or both.

In a preferred embodiment, a reflectionless band-pass electronic filtercomprising transmission lines is preferably designed as follows: First,the even-mode equivalent circuit is drawn as a terminated high-passfilter comprising a ladder network of series inductors and shuntcapacitors. It was previously shown that the transfer characteristic ofthe symmetric two-port network will be equal to the reflectioncharacteristic of the even-mode equivalent circuit. Further, uponsubstitution of transmission lines in place of the lumped elements inthe even-mode equivalent circuit, the high-pass response will beconverted to a band-stop response as a consequence of the periodicity ofthe transmission line scattering parameters. To facilitate laterapplication of identity transformations, it is useful at this stage toinsert a length of matched transmission line in cascade at the beginningof the even-mode equivalent circuit (thus affecting the reflection phaseof the circuit but not the amplitude response). The resultant even-modeequivalent circuit is thus in FIG. 1. A third-order filter is shown inthis example, but at this stage the order of the filter is arbitrary.The odd-mode equivalent circuit is preferably constructed as the dual ofthe even-mode equivalent circuit—that is, by replacing series elementswith shunt elements, shunt elements with series elements, inductors withcapacitors, and capacitors with inductors. The resistor terminations andmatched input transmission line sections remain unchanged. The resultingeven- and odd-mode equivalent circuits are shown in FIG. 2. Next, thewell-known Richard's Transformations are preferably applied to convertthe reactive elements to transmission line stubs. The result is shown inFIG. 3. Modifications are preferably made to both the even- and odd-modeequivalent circuits to restore the assumed symmetry of the filterwithout altering the port behavior, only now these modifications arebeing made to networks with transmission lines instead of only lumpedelements.

In a preferred embodiment, Kuroda's Identities are used to transform theseries short-circuited stubs on both sides to shunt open-circuited stubsspaced a quarter-wavelength apart, as shown in FIG. 4. One seriesshort-circuited stub is left un-altered at the end of the even-modeequivalent circuit.

Next, in the preferred embodiment, the positions of the remainingshort-circuited stub and the termination resistor in the even-modeequivalent circuit, which are in series, are exchanged. This leaves theshort-circuited stub now in a shunt position, as shown in FIG. 5. Aconnecting line is preferably drawn from the node between this shuntshort-circuited stub and the termination resistor in the even-modeequivalent circuit to the symmetry plane. In the odd-mode equivalentcircuit, the ground node of the termination resistor is preferablyreplaced with a virtual short on the symmetry plane, and a shuntshort-circuited stub is preferably attached to this virtual ground node.This completes restoration of the symmetry near the terminationresistors, as shown in FIG. 6.

To restore symmetry near the port nodes in a preferred embodiment, theshunt open-circuited stub at the input of the odd-mode equivalentcircuit is connected in series between the input node and the virtualground of the symmetry plane. Similarly, a series open-circuited stub ispreferably added between the input node of the even-mode equivalentcircuit and the symmetry plane. At this point, a full two-portreflectionless filter is obtained that satisfies all symmetry andduality conditions, as shown in FIG. 7. However, the seriesopen-circuited stubs are not realizable in some transmission line media.In a preferred embodiment, the transmission line identity shown in FIG.8 may be applied to remove these series open-circuited stubs. Theresultant network uses coupled transmission lines, and is shown in FIG.9.

It is instructive at this stage to contrast this topology with that of amore conventional and well-known type of absorptive filter, shown inFIG. 10. Called a directional filter, the input and output quadraturehybrids direct reflections from the two sub-filters into the terminationresistors at either end. The quadrature hybrids are often approximatedusing coupled transmission lines. This is not a reflectionless filter,however, as it only provides good impedance match over a limited rangeof frequencies where the amplitude and phase balance between thethru-port and coupled port are close to 0 dB and 90 degrees,respectively. The embodiment of the reflectionless filter in FIG. 11, onthe other hand, uses a similar set of components but is well-matched atall frequencies, including those where the amplitude and phase imbalanceof the coupled line sections is arbitrarily large.

The order of the filter chosen in the initial even-mode equivalentcircuit determines the number of open-circuited stubs in the finaltransmission line network. This number is arbitrary. In one embodiment,only a single pair of open-circuited stubs is required, as shown in FIG.11. The characteristic impedances of the transmission lines and stubsare constrained by the requirements of duality and symmetry. They may beparameterized in terms of the free parameter, x>1, as follows:

$\begin{matrix}{\rho = {{2x^{2}} - 1 + {2x\sqrt{x^{2} - 1}}}} & ( {6a} ) \\{Z_{even} = {Z_{0}\sqrt{\rho}}} & ( {6b} ) \\{Z_{odd} = {Z_{0}\text{/}\sqrt{\rho}}} & ( {6c} ) \\{Z_{oc} = \frac{Z_{0}}{x - x^{- 1}}} & ( {6d} ) \\{Z_{sc} = {Z_{0}( {x - x^{- 1}} )}} & ( {6e} ) \\{R = {Z_{0}.}} & ( {6f} )\end{matrix}$

The band-pass transfer characteristic of this circuit is shown in FIG.12. The reflection response of this circuit is identically zero at allfrequencies.

An alternate form of the reflectionless filter may be obtained by firstadding another matched transmission line segment at the input ports(equivalent to a shift in the port reference planes) prior toapplication of the transmission line identity in FIG. 8. Then theidentity is applied on these new transmission lines with the seriesstubs, essentially reversing the orientation of the coupled lines in theresultant circuit, which is shown in FIG. 13. This also leaves anadditional cascade section, denoted “Z_(x)” in the Figure. Thecharacteristic impedances are once again constrained by the requirementsof duality and symmetry, as follows:

$\begin{matrix}{\rho = {1 + {2( {x - x^{- 1}} )} + {2\sqrt{( {x - x^{- 1}} )( {1 + x - x^{- 1}} )}}}} & ( {7a} ) \\{Z_{even} = {Z_{0}\frac{2\rho}{\rho + 1}}} & ( {7b} ) \\{Z_{odd} = {Z_{0}\frac{2}{\rho + 1}}} & ( {7c} ) \\{Z_{x} = {Z_{0}x}} & ( {7d} ) \\{Z_{oc} = \frac{Z_{0}}{x - x^{- 1}}} & ( {7e} ) \\{Z_{sc} = {Z_{0}( {x - x^{- 1}} )}} & ( {7f} ) \\{R = {Z_{0}.}} & ( {7g} )\end{matrix}$

The resulting filter has exactly the same impedance and transfercharacteristics as that in FIG. 11, but with different coupled lineparameters that may be easier to fabricate in some circumstances. Asbefore, the order of the filter and the consequent number oftransmission line stubs is arbitrary.

In the previous embodiments, the filter resonators were formed bytransmission-line stubs. In other embodiments, one or more of thetransmission line stubs may be replaced by additional cascadedtransmission lines. In a preferred embodiment, these additional cascadedtransmission lines have characteristic impedance given by

Z _(r) =Z ₀ x ⁻¹  (8)

An example is shown in FIG. 14, obtained by replacing the stubresonators, Z_(oc), in FIG. 11 with a line resonator, Z_(r). Itssimulated performance is shown in FIG. 15. In general the cascaded lineresonators will result in lower side lobes, at the cost of more roundedpassband corners.

Other embodiments and uses of the invention will be apparent to thoseskilled in the art from consideration of the specification and practiceof the invention disclosed herein. All references cited herein,including all publications, U.S. and foreign patents and patentapplications, are specifically and entirely incorporated by reference.It is intended that the specification and examples be consideredexemplary only with the true scope and spirit of the invention indicatedby the following claims. Furthermore, the term “comprising of” includesthe terms “consisting of” and “consisting essentially of.”

1. A reflectionless, impedance-matched, electronic filter comprising: asymmetric two-port network, wherein the symmetry defines an even-modeequivalent circuit and an odd-mode equivalent circuit when the ports aredriven in-phase and 180-degrees out-of-phase, respectively, such that: anormalized input impedance of the even-mode equivalent circuit issubstantially equal to the normalized input admittance of the odd-modeequivalent circuit; and a normalized input admittance of the even-modeequivalent circuit is substantially equal to the normalized inputimpedance of the odd-mode equivalent circuit; wherein the two ports areknown as the input and output, respectively, wherein the ports areconnected through a pair of coupled-line sections, one on the input-portside, the other on the output-port side, wherein the coupledtransmission-line sections connected to the ports are also connected toidentical chains of cascaded transmission-line sections,transmission-line stubs, and additional coupled transmission lines, withone chain on the input side and one chain on the output side; whereinthe chains on each side are terminated by resistors or one or moreimpedance-matched internal subnetworks; wherein the resistors or matchedinternal subnetworks may be grounded through transmission-line stubs. 2.The reflectionless filter of claim 1, wherein the chains on the inputside and the output side may be connected at one or more points bycoupled transmission lines.
 3. The reflectionless filter of claim 1,wherein the transmission lines, stubs, and coupled lines are allessentially a quarter wavelength long at the center frequency ofoperation.
 4. The reflectionless filter of claim 1, wherein theidentical chains on the input side and the output side each comprise analternating pattern of cascade transmission lines and open-circuitedstubs.
 5. The reflectionless filter of claim 4, wherein the normalizedcharacteristic impedance of the cascade lines in each of the identicalchains on the input side and output side is labeled x; and wherein thenormalized characteristic impedance of the open-circuited stubs in theidentical chains is (x−x⁻¹)⁻¹; and wherein the resistors or matchedinternal subnetworks are grounded through transmission-line stubs havingnormalized characteristic impedance (x−x⁻¹);
 6. The reflectionlessfilter of claim 5, wherein the coupled lines connected to the input portand the output port have coupling factorρ=1+2(x−x⁻¹)+2[(x−x⁻¹)(1+x−x⁻¹)]^(1/2); and wherein the normalizedeven-mode characteristic impedance of the coupled lines is 2ρ/(ρ+1) andthe normalized odd-mode characteristic impedance of the couple lines is2/(ρ+1).
 7. The reflectionless filter of claim 5, wherein the coupledlines connected to the input port and the output port have couplingfactor ρ=2x²−1+2x(x²−1)^(1/2); and wherein the normalized even-modecharacteristic impedance of the coupled lines is ρ^(1/2) and thenormalized odd-mode characteristic impedance of the couple lines isρ^(−1/2).
 8. The reflectionless filter of claim 1, wherein the identicalchains on the input side and the output side each comprise a pattern ofcascade transmission lines of alternating high- and low-impedance,relative to one another.
 9. The reflectionless filter of claim 8,wherein the cascade transmission lines in each chain with relativelyhigh impedance have normalized characteristic impedance labeled x, andthe cascade transmission lines in each chain with relatively lowimpedance of normalized characteristic impedance x⁻¹; and wherein theresistors or matched internal subnetworks are grounded throughtransmission-line stubs having normalized characteristic impedance(x−x⁻¹);
 10. The reflectionless filter of claim 9, wherein the coupledlines connected to the input port and the output port have couplingfactor ρ=1+2(x−x⁻¹)+2[(x−x⁻¹)(1+x−x⁻¹)]^(1/2); and wherein thenormalized even-mode characteristic impedance of the coupled lines is2ρ/(ρ+1) and the normalized odd-mode characteristic impedance of thecouple lines is 2/(ρ+1).
 11. The reflectionless filter of claim 9,wherein the coupled lines connected to the input port and the outputport have coupling factor ρ=2x²−1+2x(x²-1)^(1/2); and wherein thenormalized even-mode characteristic impedance of the coupled lines isρ^(1/2) and the normalized odd-mode characteristic impedance of thecouple lines is ρ^(1/2).
 12. The reflectionless filter of claim 1,wherein the resistors or matched internal subnetworks are groundedthrough matched transmission-line stubs which are short-circuited. 13.The reflectionless filter of claim 1, wherein the even- and odd-modeequivalent circuits exhibit a low-pass response, and the completedtwo-port filter exhibits a high-pass or band-pass response.
 14. Thereflectionless filter of claim 1, wherein the identical chains on theinput side and the output side are terminated by an impedance-matchedinternal subnetwork which connects to the two chains.
 15. Thereflectionless filter of claim 14, wherein the impedance-matchedinternal subnetwork comprises one or more of transmission lines, lumpedelements, and active circuits.